Expand all Collapse all | Results 101 - 125 of 217 |
101. CJM 2007 (vol 59 pp. 1050)
On the Restriction to $\D^* \times \D^*$ of Representations of $p$-Adic $\GL_2(\D)$ Let $\mathcal{D}$ be a division algebra
over a nonarchimedean local field. Given
an irreducible representation $\pi$ of $\GL_2(\mathcal{D})$, we
describe its restriction to the diagonal subgroup $\mathcal{D}^* \times
\mathcal{D}^*$. The description is in terms of the structure of the
twisted Jacquet module of the representation $\pi$. The proof
involves Kirillov theory that we have developed earlier in joint work
with Dipendra Prasad. The main result on restriction also shows that
$\pi$ is $\mathcal{D}^* \times \mathcal{D}^*$-distinguished if and only if
$\pi$ admits a Shalika model. We further prove that if $\mathcal{D}$
is a quaternion division algebra then the twisted Jacquet module
is multiplicity-free by proving an appropriate theorem on invariant
distributions; this then proves a multiplicity-one theorem on the
restriction to $\mathcal{D}^* \times \mathcal{D}^*$ in the quaternionic
case.
Categories:22E50, 22E35, 11S37 |
102. CJM 2007 (vol 59 pp. 673)
Hecke $L$-Functions and the Distribution of Totally Positive Integers Let $K$ be a totally real number field of degree $n$. We show that
the number of totally positive integers
(or more generally the number of totally positive elements of a given fractional ideal)
of given trace is evenly distributed around its expected value, which is
obtained from geometric considerations.
This result depends on unfolding an integral over
a compact torus.
Keywords:Eisenstein series, toroidal integral, Fourier series, Hecke $L$-function, totally positive integer, trace Categories:11M41, 11F30, , 11F55, 11H06, 11R47 |
103. CJM 2007 (vol 59 pp. 553)
Computations of Elliptic Units for Real Quadratic Fields Let $K$ be a real quadratic field, and $p$ a rational prime which is
inert in $K$. Let $\alpha$ be a modular unit on $\Gamma_0(N)$. In an
earlier joint article with Henri Darmon, we presented the definition
of an element $u(\alpha, \tau) \in K_p^\times$ attached to $\alpha$
and each $\tau \in K$. We conjectured that the $p$-adic number
$u(\alpha, \tau)$ lies in a specific ring class extension of $K$
depending on $\tau$, and proposed a ``Shimura reciprocity law"
describing the permutation action of Galois on the set of $u(\alpha,
\tau)$. This article provides computational evidence for these
conjectures. We present an efficient algorithm for computing
$u(\alpha, \tau)$, and implement this algorithm with the modular unit
$\alpha(z) = \Delta(z)^2\Delta(4z)/\Delta(2z)^3.$ Using $p = 3, 5, 7,$
and $11$, and all real quadratic fields $K$ with discriminant $D <
500$ such that $2$ splits in $K$ and $K$ contains no unit of negative
norm, we obtain results supporting our conjectures. One of the
theoretical results in this paper is that a certain measure used to
define $u(\alpha, \tau)$ is shown to be $\mathbf{Z}$-valued rather
than only $\mathbf{Z}_p \cap \mathbf{Q}$-valued; this is an
improvement over our previous result and allows for a precise
definition of $u(\alpha, \tau)$, instead of only up to a root of
unity.
Categories:11R37, 11R11, 11Y40 |
104. CJM 2007 (vol 59 pp. 503)
Cyclic Groups and the Three Distance Theorem We give a two dimensional extension of the three distance Theorem. Let
$\theta$ be in $\mathbf{R}^{2}$ and let $q$ be in $\mathbf{N}$. There exists a
triangulation of $\mathbf{R}^{2}$ invariant by $\mathbf{Z}^{2}$-translations,
whose set of vertices is $\mathbf{Z}^{2}+\{0,\theta,\dots,q\theta\}$, and whose
number of different triangles, up to translations, is bounded above by a
constant which does not depend on $\theta$ and $q$.
Categories:11J70, 11J71, 11J13 |
105. CJM 2007 (vol 59 pp. 372)
Zeta Functions of Supersingular Curves of Genus 2 We determine which isogeny classes of supersingular abelian
surfaces over a finite field $k$ of characteristic $2$ contain
jacobians. We deal with this problem in a direct way by computing
explicitly the zeta function of all supersingular curves of genus
$2$. Our procedure is constructive, so that we are able to exhibit
curves with prescribed zeta function and find formulas for the
number of curves, up to $k$-isomorphism, leading to the same zeta
function.
Categories:11G20, 14G15, 11G10 |
106. CJM 2007 (vol 59 pp. 85)
On the Convergence of a Class of Nearly Alternating Series Let $C$ be the class of convex sequences of real numbers. The
quadratic irrational numbers can be partitioned into two types as
follows. If $\alpha$ is of the first type and $(c_k) \in C$, then
$\sum (-1)^{\lfloor k\alpha \rfloor} c_k$ converges if and only if
$c_k \log k \rightarrow 0$. If $\alpha$ is of the second type and
$(c_k) \in C$, then $\sum (-1)^{\lfloor k\alpha \rfloor} c_k$
converges if and only if $\sum c_k/k$ converges. An example of a
quadratic irrational of the first type is $\sqrt{2}$, and an
example of the second type is $\sqrt{3}$. The analysis of this
problem relies heavily on the representation of $ \alpha$ as a
simple continued fraction and on properties of the sequences of
partial sums $S(n)=\sum_{k=1}^n (-1)^{\lfloor k\alpha \rfloor}$
and double partial sums $T(n)=\sum_{k=1}^n S(k)$.
Keywords:Series, convergence, almost alternating, convex, continued fractions Categories:40A05, 11A55, 11B83 |
107. CJM 2007 (vol 59 pp. 148)
On Certain Classes of Unitary Representations for Split Classical Groups In this paper we prove the unitarity of duals of tempered
representations supported on minimal parabolic subgroups for split
classical $p$-adic groups. We also construct a family of unitary
spherical representations for real and complex classical groups
Categories:22E35, 22E50, 11F70 |
108. CJM 2007 (vol 59 pp. 211)
On Two Exponents of Approximation Related to a Real Number and Its Square For each real number $\xi$, let $\lambdahat_2(\xi)$ denote the
supremum of all real numbers $\lambda$ such that, for each
sufficiently large $X$, the inequalities $|x_0| \le X$,
$|x_0\xi-x_1| \le X^{-\lambda}$ and $|x_0\xi^2-x_2| \le
X^{-\lambda}$ admit a solution in integers $x_0$, $x_1$ and $x_2$
not all zero, and let $\omegahat_2(\xi)$ denote the supremum of
all real numbers $\omega$ such that, for each sufficiently large
$X$, the dual inequalities $|x_0+x_1\xi+x_2\xi^2| \le
X^{-\omega}$, $|x_1| \le X$ and $|x_2| \le X$ admit a solution in
integers $x_0$, $x_1$ and $x_2$ not all zero. Answering a
question of Y.~Bugeaud and M.~Laurent, we show that the exponents
$\lambdahat_2(\xi)$ where $\xi$ ranges through all real numbers
with $[\bQ(\xi)\wcol\bQ]>2$ form a dense subset of the interval $[1/2,
(\sqrt{5}-1)/2]$ while, for the same values of $\xi$, the dual
exponents $\omegahat_2(\xi)$ form a dense subset of $[2,
(\sqrt{5}+3)/2]$. Part of the proof rests on a result of
V.~Jarn\'{\i}k showing that $\lambdahat_2(\xi) =
1-\omegahat_2(\xi)^{-1}$ for any real number $\xi$ with
$[\bQ(\xi)\wcol\bQ]>2$.
Categories:11J13, 11J82 |
109. CJM 2007 (vol 59 pp. 127)
Smooth Values of the Iterates of the Euler Phi-Function Let $\phi(n)$ be the Euler phi-function, define
$\phi_0(n) = n$ and $\phi_{k+1}(n)=\phi(\phi_{k}(n))$ for all
$k\geq 0$. We will determine an asymptotic formula for the set of
integers $n$ less than $x$ for which $\phi_k(n)$ is $y$-smooth,
conditionally on a weak form of the Elliott--Halberstam conjecture.
Categories:11N37, 11B37, 34K05, 45J05 |
110. CJM 2006 (vol 58 pp. 1203)
Orbites unipotentes et pÃ´les d'ordre maximal de la fonction $\mu $ de Harish-Chandra Dans un travail ant\'erieur, nous
avions montr\'e que l'induite parabolique (normalis\'ee) d'une
repr\'esentation irr\'eductible cuspidale $\sigma $ d'un
sous-groupe de Levi $M$ d'un groupe $p$-adique contient un
sous-quotient de carr\'e int\'egrable, si et seulement si la
fonction $\mu $ de Harish-Chandra a un p\^ole en $\sigma $ d'ordre
\'egal au rang parabolique de $M$. L'objet de cet article est
d'interpr\'eter ce r\'esultat en termes de fonctorialit\'e de
Langlands.
Categories:11F70, 11F80, 22E50 |
111. CJM 2006 (vol 58 pp. 1095)
A Casselman--Shalika Formula for the Shalika Model of $\operatorname{GL}_n$ The Casselman--Shalika method is a way to compute explicit
formulas for periods of irreducible unramified representations of
$p$-adic groups that are associated to unique models (i.e.,
multiplicity-free induced representations). We apply this method
to the case of the Shalika model of $GL_n$, which is known to
distinguish lifts from odd orthogonal groups. In the course of our
proof, we further develop a variant of the method, that was
introduced by Y. Hironaka, and in effect reduce many such problems
to straightforward calculations on the group.
Keywords:Casselman--Shalika, periods, Shalika model, spherical functions, Gelfand pairs Categories:22E50, 11F70, 11F85 |
112. CJM 2006 (vol 58 pp. 796)
Mordell--Weil Groups and the Rank of Elliptic Curves over Large Fields Let $K$ be a number field, $\overline{K}$ an algebraic closure of
$K$ and $E/K$ an elliptic curve
defined over $K$. In this paper, we prove that if $E/K$ has a
$K$-rational point $P$ such that $2P\neq O$ and $3P\neq O$, then
for each $\sigma\in \Gal(\overline{K}/K)$, the Mordell--Weil group
$E(\overline{K}^{\sigma})$ of $E$ over the fixed subfield of
$\overline{K}$ under $\sigma$ has infinite rank.
Category:11G05 |
113. CJM 2006 (vol 58 pp. 843)
On the One-Level Density Conjecture for Quadratic Dirichlet L-Functions In a previous article, we studied the distribution of ``low-lying"
zeros of the family of quad\-ratic Dirichlet $L$-functions assuming
the Generalized Riemann Hypothesis for all Dirichlet
$L$-functions. Even with this very strong assumption, we were
limited to using weight functions whose Fourier transforms are
supported in the interval $(-2,2)$. However, it is widely believed
that this restriction may be removed, and this leads to what has
become known as the One-Level Density Conjecture for the zeros of
this family of quadratic $L$-functions. In this note, we make use
of Weil's explicit formula as modified by Besenfelder to prove an
analogue of this conjecture.
Category:11M26 |
114. CJM 2006 (vol 58 pp. 643)
Centralizers and Twisted Centralizers: Application to Intertwining Operators ABSTRACT
The equality of the centralizer and twisted centralizer is proved
based on a case-by-case analysis when the unipotent radical of a
maximal parabolic subgroup is abelian.
Then this result is used to determine the poles of intertwining operators.
Category:11F70 |
115. CJM 2006 (vol 58 pp. 580)
Annihilators for the Class Group of a Cyclic Field of Prime Power Degree, II We prove, for a field $K$ which is cyclic of odd prime power
degree over the rationals, that the annihilator of the
quotient of the units of $K$ by a suitable large subgroup (constructed
from circular units) annihilates what we call the
non-genus part of the class group.
This leads to stronger annihilation results for the whole
class group than a routine application of the Rubin--Thaine method
would produce, since the
part of the class group determined by genus theory has an obvious
large annihilator which is not detected by
that method; this is our reason for concentrating on
the non-genus part. The present work builds on and strengthens
previous work of the authors; the proofs are more conceptual now,
and we are also able to construct an example which demonstrates
that our results cannot be easily sharpened further.
Categories:11R33, 11R20, 11Y40 |
116. CJM 2006 (vol 58 pp. 419)
Stark's Conjecture and New Stickelberger Phenomena We introduce a new conjecture concerning the construction
of elements in the annihilator ideal
associated to a Galois action on the higher-dimensional algebraic
$K$-groups of rings of integers in number fields. Our conjecture is
motivic in the sense that it involves the (transcendental) Borel
regulator as well as being related to $l$-adic \'{e}tale
cohomology. In addition, the conjecture generalises the well-known
Coates--Sinnott conjecture. For example, for a totally real
extension when $r = -2, -4, -6, \dotsc$ the Coates--Sinnott
conjecture merely predicts that zero annihilates $K_{-2r}$ of the
ring of $S$-integers while our conjecture predicts a non-trivial
annihilator. By way of supporting evidence, we prove the
corresponding (conjecturally equivalent) conjecture for the Galois
action on the \'{e}tale cohomology of the cyclotomic extensions of
the rationals.
Categories:11G55, 11R34, 11R42, 19F27 |
117. CJM 2006 (vol 58 pp. 3)
The Functional Equation of Zeta Distributions Associated With Non-Euclidean Jordan Algebras This paper is devoted to the study of certain zeta distributions
associated with simple non-Euclidean Jordan algebras. An explicit
form of the corresponding functional equation and Bernstein-type
identities is obtained.
Keywords:Zeta distributions, functional equations, Bernstein polynomials, non-Euclidean Jordan algebras Categories:11M41, 17C20, 11S90 |
118. CJM 2006 (vol 58 pp. 115)
Quelques rÃ©sultats sur les Ã©quations $ax^p+by^p=cz^2$ Let $p$ be a prime number $\geq 5$ and $a,b,c$ be non
zero natural numbers. Using the works of K. Ribet and A. Wiles on the
modular representations, we get new results about the description of
the primitive solutions of the diophantine equation $ax^p+by^p=cz^2$,
in case the product of the prime divisors of $abc$ divides $2\ell$,
with $\ell$ an odd prime number. For instance, under some conditions
on $a,b,c$, we provide a constant $f(a,b,c)$ such that there are no
such solutions if $p>f(a,b,c)$. In application, we obtain information
concerning the $\Q$-rational points of hyperelliptic curves given by
the equation $y^2=x^p+d$ with $d\in \Z$.
Category:11G |
119. CJM 2005 (vol 57 pp. 1155)
The Square Sieve and the Lang--Trotter Conjecture Let $E$ be an elliptic curve defined over $\Q$ and without
complex multiplication. Let $K$ be a fixed imaginary quadratic field.
We find nontrivial upper bounds for the number of ordinary primes $p \leq x$
for which $\Q(\pi_p) = K$, where $\pi_p$ denotes the Frobenius endomorphism
of $E$ at $p$. More precisely, under a generalized Riemann hypothesis
we show that this number is $O_{E}(x^{\slfrac{17}{18}}\log x)$, and unconditionally
we show that this number is $O_{E, K}\bigl(\frac{x(\log \log x)^{\slfrac{13}{12}}}
{(\log x)^{\slfrac{25}{24}}}\bigr)$. We also prove that the number of imaginary quadratic
fields $K$, with $-\disc K \leq x$ and of the form $K = \Q(\pi_p)$, is
$\gg_E\log\log\log x$ for $x\geq x_0(E)$. These results represent progress towards
a 1976 Lang--Trotter conjecture.
Keywords:Elliptic curves modulo $p$; Lang--Trotter conjecture;, applications of sieve methods Categories:11G05, 11N36, 11R45 |
120. CJM 2005 (vol 57 pp. 1215)
Reciprocity Law for Compatible Systems of Abelian $\bmod p$ Galois Representations The main result of the paper
is a {\em reciprocity law} which proves that
compatible systems of semisimple, abelian mod $p$ representations
(of arbitrary dimension)
of absolute Galois groups of number fields, arise from Hecke characters.
In the last section analogs for Galois groups of function fields of these
results are explored, and a question is raised whose answer seems to
require developments in transcendence theory in characteristic $p$.
Category:11F80 |
121. CJM 2005 (vol 57 pp. 1102)
Power Residues of Fourier Coefficients of Modular Forms Let $\rho \colon G_{\Q} \to \GL_{n}(\Ql)$ be a motivic $\ell$-adic Galois
representation. For fixed $m > 1$ we initiate an investigation of the
density of the set of primes $p$ such that the trace of the image of an
arithmetic Frobenius at $p$ under $\rho$ is an $m$-th power residue
modulo $p$. Based on numerical investigations with modular forms we
conjecture (with Ramakrishna) that this density equals $1/m$ whenever the
image of $\rho$ is open. We further conjecture that for such $\rho$ the set
of these primes $p$ is independent of any set defined by Cebatorev-style
Galois-theoretic conditions (in an appropriate sense). We then compute these
densities for certain $m$ in the complementary case of modular forms of
CM-type with rational Fourier coefficients; our proofs are a combination of
the Cebatorev density theorem (which does apply in the CM case) and
reciprocity laws applied to Hecke characters. We also discuss a potential
application (suggested by Ramakrishna) to computing inertial degrees at $p$
in abelian extensions of imaginary quadratic fields unramified away from $p$.
Categories:11F30, 11G15, 11A15 |
122. CJM 2005 (vol 57 pp. 1080)
The Gelfond--Schnirelman Method in Prime Number Theory The original Gelfond--Schnirelman method, proposed in 1936, uses
polynomials with integer coefficients and small norms on $[0,1]$
to give a Chebyshev-type lower bound in prime number theory. We
study a generalization of this method for polynomials in many
variables. Our main result is a lower bound for the integral of
Chebyshev's $\psi$-function, expressed in terms of the weighted
capacity. This extends previous work of Nair and Chudnovsky, and
connects the subject to the potential theory with external fields
generated by polynomial-type weights. We also solve the
corresponding potential theoretic problem, by finding the extremal
measure and its support.
Keywords:distribution of prime numbers, polynomials, integer, coefficients, weighted transfinite diameter, weighted capacity, potentials Categories:11N05, 31A15, 11C08 |
123. CJM 2005 (vol 57 pp. 812)
On the Vanishing of $\mu$-Invariants of Elliptic Curves over $\qq$ Let $E_{/\qq}$ be an elliptic curve with good ordinary reduction at a
prime $p>2$. It has a well-defined Iwasawa $\mu$-invariant $\mu(E)_p$
which encodes part of the information about the growth of the Selmer
group $\sel E{K_n}$ as $K_n$ ranges over the subfields of the
cyclotomic $\zzp$-extension $K_\infty/\qq$. Ralph Greenberg has
conjectured that any such $E$ is isogenous to a curve $E'$ with
$\mu(E')_p=0$. In this paper we prove Greenberg's conjecture for
infinitely many curves $E$ with a rational $p$-torsion point, $p=3$ or
$5$, no two of our examples having isomorphic $p$-torsion. The core
of our strategy is a partial explicit evaluation of the global duality
pairing for finite flat group schemes over rings of integers.
Category:11R23 |
124. CJM 2005 (vol 57 pp. 494)
Summation Formulae for Coefficients of $L$-functions With applications in mind we establish a summation formula for the
coefficients of a general Dirichlet series satisfying a suitable
functional equation. Among a number of consequences we derive a
generalization of an elegant divisor sum bound due to F.~V. Atkinson.
Categories:11M06, 11M41 |
125. CJM 2005 (vol 57 pp. 535)
On Local $L$-Functions and Normalized Intertwining Operators In this paper we make explicit all $L$-functions in the
Langlands--Shahidi method which appear as normalizing factors of
global intertwining operators in the constant term of the
Eisenstein series. We prove, in many cases,
the conjecture of Shahidi regarding the
holomorphy of the local $L$-functions. We also prove
that the normalized local intertwining operators are holomorphic and
non-vaninishing for $\re(s)\geq 1/2$ in many cases. These local
results are essential in global applications such as Langlands
functoriality, residual spectrum and determining poles of
automorphic $L$-functions.
Categories:11F70, 22E55 |